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LIST EDGE AND LIST TOTAL COLORINGS OF PLANAR GRAPHS WITHOUT 6-CYCLES WITH CHORD
2012
Bulletin of the Korean Mathematical Society
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It is proved that if a planar graph G without 6-cycles with chord, then χ ′ l ...
Giving a planar graph G, let χ ′ l (G) and χ ′′ l (G) denote the list edge chromatic number and list total chromatic number of G respectively. ...
Planar graphs without 6-cycles with chord First let us introduce an important lemma. Proof. For G is a critical planar graph with ∆(G) ≥ 6, then G contains no (4, 4, 5 − )-face f = uvw. ...
doi:10.4134/bkms.2012.49.2.359
fatcat:oxtudhudpndhpd2d6sk7ffgoea
3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles
2010
SIAM Journal on Discrete Mathematics
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A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. ...
In addition, this implies that every triangle-free planar graph without 6-and 7-cycles is 3-choosable. * Supported by a CZ-SL bilateral project MEB 090805 and BI-CZ/08-09-005. ...
This strengthens the results of Lidický [6] that planar graphs without 3-, 6-, 7-and 8-cycles are 3-choosable, and of Zhang and Xu [13] that planar graphs without 3-, 6-, 7-and 9-cycles are 3-choosable ...
doi:10.1137/080743020
fatcat:qq6666y2b5bk5f5kw3reox6rbu
Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8
[article]
2016
arXiv
pre-print
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Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin. ...
We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring. ...
On the other hand, planar graphs are 5-choosable [14] , and every planar graph without cycles of lengths 3 and 4 is 3-choosable [15] . ...
arXiv:1508.03437v2
fatcat:iqjnicgb5vcthd2r3flfhifai4
List Edge Colorings of Planar Graphs with 7-cycles Containing at Most Two Chords
2019
Applied Science and Innovative Research
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In this paper we prove that if G is a planar graph, and each 7-cycle contains at most two chords, then G is edge-k-choosable, where k = max{8, ?(G) + 1}. ...
Structural Properties of Planar Graphs with 7-Cycles Containing at Most Two Chords Lemma 4. Let G be a planar graph in which each 7-cycle contains at most two chords. ...
Suppose that G is a counterexample to our theorem with the minimum number of edges and G is any planar graph in which every 7-cycle contains at most two chords. ...
doi:10.22158/asir.v3n2p85
fatcat:n5lr2zt5ungkri6v7de2edl6em
On oriented cliques with respect to push operation
[article]
2015
arXiv
pre-print
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To push a vertex v of a directed graph G is to change the orientations of all the arcs incident with v. An oriented graph is a directed graph without any cycle of length at most 2. ...
We also provide an exhaustive list of minimal (with respect to spanning subgraph inclusion) planar push cliques. ...
Note that an eight cycle can have three types of chords, namely, a very long chord connecting vertices at distance 4, a long chord and a short chord. ...
arXiv:1511.08672v1
fatcat:gayvb57bpngojgdr77mazhrqra
On Choosability with Separation of Planar Graphs with Forbidden Cycles
2015
Journal of Graph Theory
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We prove that planar graphs without 4-cycles are (3,1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3,1)-choosable. ...
We study choosability with separation which is a constrained version of list coloring of graphs. ...
Jao for fruitful discussions and encouragement in the early stage of the project. ...
doi:10.1002/jgt.21875
fatcat:5z7apx6gsredpmlpgdu2gujdn4
Algorithms
[chapter]
2011
Graph Coloring Problems
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Graphs
97
5.1
Critical Graphs With Many Edges
97
5.2
Minimum Degree of 4-and 5-Critical Graphs
>
98
5.3
Critical Graphs With Few Edges
99
5.4
Four-Critical Amenable Graphs
101
5.5
Four-Critical ...
of Hamilton Cycles
.
82
4.6
Brooks' Theorem for Triangle-Free Graphs
83
4.7
Graphs Without Large Complete Subgraphs
85
4.8
^-Chromatic Graphs of Maximum Degree k
85
4.9
Total Coloring
86 ...
doi:10.1002/9781118032497.ch10
fatcat:374tktuvgvekni4fnz3dgbytjm
Do triangle-free planar graphs have exponentially many 3-colorings?
[article]
2017
arXiv
pre-print
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Thomassen conjectured that triangle-free planar graphs have an exponential number of 3-colorings. ...
We show this conjecture to be equivalent to the following statement: there exists a positive real α such that whenever G is a planar graph and A is a subset of its edges whose deletion makes G triangle-free ...
coloring of planar graphs. ...
arXiv:1702.00588v2
fatcat:xv4etv6fkfaorkxvpkzg2qdv4q
Planar graphs without cycles of length 4, 7, 8, or 9 are 3-choosable
2011
Discrete Applied Mathematics
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It is known that planar graphs without cycles of length 4, i, j, or 9 with 4 < i < j < 9, except that i = 7 and j = 8, are 3-choosable. ...
This paper proves that planar graphs without cycles of length 4, 7, 8, or 9 are also 3-choosable. ...
Acknowledgements The authors are grateful to the referees for their wise suggestions to simplify the original proof of this paper. ...
doi:10.1016/j.dam.2010.11.002
fatcat:g6hk5nmh6rhjnaefrrxjcnwmlm
Improper choosability of graphs of nonnegative characteristic
2008
Computers and Mathematics with Applications
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In this paper, we prove: (1) If G contains no k-cycle with a chord for all k = 4, 5, 6, then G is (3, 1) *choosable; (2) If G contains neither 5-cycle with a chord nor 6-cycle with a chord, then G is ( ...
A graph G is called (k, d) * -choosable if, for every list assignment L with |L(v)| = k for all v ∈ V (G), there is an L-coloring of G such that every vertex has at most d neighbors having the same color ...
Equivalently, every torodial graph without a 4-cycle with a chord is (4, 1) * -choosable. Theorems 5 and 6 are, to some extent, an extension to their result. ...
doi:10.1016/j.camwa.2008.03.036
fatcat:437w5yexyrdplp74g6fmfvu3g4
List coloring of planar graphs without 6-cycles with two chords
2018
Discussiones Mathematicae Graph Theory
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If ∆(G) ≥ 6, then χ ′ list (G) ≤ ∆(G) + 1. The following result is about edge-∆-choosable of embedded planar graphs without 6-cycles with two chords. Theorem 8. ...
Main Results and Their Proofs In the section, we always assume that all graphs are planar graphs that have been embedded in the plane and G is a planar graph without 6-cycles with two chords. ...
By Lemmas 9 and 10, v may be the 3master of two 3 − -vertices and the 2-master of a 2-vertex, that is, v sends at most 5 to its incident 3 − -vertices. Suppose that d(v) = 9. Then f 3 (v) ≤ 6. ...
doi:10.7151/dmgt.2183
fatcat:le3o6s6lzna35ebvqzdgudd57i
(4, 2)-Choosability of Planar Graphs with Forbidden Structures
2017
Graphs and Combinatorics
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A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ∈{5,6,7} that a planar graph is (4,2)-choosable if it does not contain chorded ℓ-cycles. ...
A graph is (k,s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. ...
Conclusion and Future Work We proved that, for each k ∈ {5, 6, 7}, planar graphs with no chorded k-cycles are (4, 2)-choosable. ...
doi:10.1007/s00373-017-1812-5
fatcat:ta5azl3h7nhhfoqnlgnmjmyswm
On oriented cliques with respect to push operation
2017
Discrete Applied Mathematics
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An oriented graph is a directed graph without any directed cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. ...
We also prove that a planar push clique can have at most 8 vertices and provide an exhaustive list of planar push cliques. ...
Acknowledgement: The authors would like to thank the anonymous reviewer for the constructive comments towards improvement of the content, clarity and conciseness of the manuscript. ...
doi:10.1016/j.dam.2017.07.037
fatcat:na4wiolki5gp3eq2653zyy73ne
Planar graphs without normally adjacent short cycles
[article]
2021
arXiv
pre-print
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Consequently, every planar graph without 4-, 6-, 8-cycles is 3-choosable, and every planar graph without 4-, 5-, 7-, 8-cycles is 3-choosable. ...
Let 𝒢 be the class of plane graphs without triangles normally adjacent to 8^--cycles, without 4-cycles normally adjacent to 6^--cycles, and without normally adjacent 5-cycles. ...
This work was supported by the National Natural Science Foundation of China and partially supported by the Fundamental Research Funds for Universities in Henan (YQPY20140051). ...
arXiv:1908.04902v3
fatcat:fho4mz3b5vdpldbq2eobl6jgqi
On choosability with separation of planar graphs with lists of different sizes
2015
Discrete Mathematics
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A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. ...
A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. ...
Choi et. al [1] proved that every planar graph without 4-cycles is (3, 1)-choosable and that every planar graph without 5-cycles and 6-cycles is (3, 1)-choosable. ...
doi:10.1016/j.disc.2015.01.008
fatcat:njxw7lf4j5fijlqswvwnwg6xea
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